Chapter 9 - Further Applications of the Integral and Taylor Polynomials - 9.1 Arc Length and Surface Area - Preliminary Questions - Page 468: 5

$$ \int_{1}^{4} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x \geq 3 $$

Since $f^{\prime}(x)^{2} \geq 0,$ we know that $\sqrt{1+\left[f^{\prime}(x)\right]^{2}} \geq \sqrt{1}=1.$ Then the arc length of the graph of $f(x)$ on $[1,4]$ is $$ \int_{1}^{4} \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x \geq \int_{1}^{4} 1 d x=3 $$